Optimal probabilistic dense coding schemes

Abstract

Dense coding with non-maximally entangled states has been investigated in many different scenarios. We revisit this problem for protocols adopting the standard encoding scheme. In this case, the set of possible classical messages cannot be perfectly distinguished due to the non-orthogonality of the quantum states carrying them. So far, the decoding process has been approached in two ways: (i) The message is always inferred, but with an associated (minimum) error; (ii) the message is inferred without error, but only sometimes; in case of failure, nothing else is done. Here, we generalize on these approaches and propose novel optimal probabilistic decoding schemes. The first uses quantum-state separation to increase the distinguishability of the messages with an optimal success probability. This scheme is shown to include (i) and (ii) as special cases and continuously interpolate between them, which enables the decoder to trade-off between the level of confidence desired to identify the received messages and the success probability for doing so. The second scheme, called multistage decoding, applies only for qudits (d-level quantum systems with $d>2$) and consists of further attempts in the state identification process in case of failure in the first one. We show that this scheme is advantageous over (ii) as it increases the mutual information between the sender and receiver.

Appendix

In this appendix we show that the mutual information between Alice and Bob in the dense coding protocol described in Sect. 2 is given by Eqs. (11) and (12). For a pair of random variables M and R, the mutual information is defined as [32]

$$\begin{aligned} I(M;R)=H(M)-H(M|R), \end{aligned}$$

(36)

where H(M) and H(M|R) are the Shannon and conditional entropies, respectively. In the present context, the random variable M is associated with the set of possible messages $\{(j,k)|j=0,\ldots ,D-1;k=0,\ldots ,d_2-1\}$ encoded by Alice in the quantum state $\hat{\rho }_{jk}=|\alpha _j\rangle \langle \alpha _j|\otimes |k\rangle \langle k|$ [see Eq. (10)] with a probability $p(\hat{\rho }_{jk})=1/(d_2D)$. Thus, the Shannon entropy will be given by

$$\begin{aligned} H(M)= & {} -\sum _{j=0}^{D-1}\sum _{k=0}^{d_2-1}p(\hat{\rho }_{jk})\log _2p(\hat{\rho }_{jk}) \nonumber \\= & {} \sum _{j=0}^{D-1}\sum _{k=0}^{d_2-1}\frac{1}{d_2D}\log _2\frac{1}{d_2D}\nonumber \\= & {} \log _2d_2D. \end{aligned}$$

(37)

On the other hand, the random variable R is associated with the set of Bob’s measurement results $\{\omega _{lm}|l=0,\ldots ,N-1;m=0,\ldots ,d_2-1\}$ in his decoding process. The index l (m) is connected with the result of a measurement on system 1 (2), and the number of values it assumes, N ($d_2$), will be explained later. With these definitions, the conditional entropy will be written as

$$\begin{aligned} H(M|R)=-\sum _{j=0}^{D-1}\sum _{l=0}^{N-1}\sum _{k,m=0}^{d_2-1}p(\omega _{lm}) p(\hat{\rho }_{jk}|\omega _{lm})\log _2p(\hat{\rho }_{jk}|\omega _{lm}). \end{aligned}$$

(38)

Using Bayes’ rule

$$\begin{aligned} p(\hat{\rho }_{jk}|\omega _{lm})=p(\omega _{lm}|\hat{\rho }_{jk})\frac{p(\hat{\rho }_{jk})}{p(\omega _{lm})}, \end{aligned}$$

(39)

we can rewrite (38) as

$$\begin{aligned} H(M|R)=-\sum _{j=0}^{D-1}\sum _{l=0}^{N-1}\sum _{k,m=0}^{d_2-1}p(\hat{\rho }_{jk}) p(\omega _{lm}|\hat{\rho }_{jk})\log _2\left[ p(\omega _{lm}|\hat{\rho }_{jk}) \frac{p(\hat{\rho }_{jk})}{p(\omega _{lm})}\right] , \end{aligned}$$

(40)

where $p(\omega _{lm})$ is the overall probability of obtaining the outcome $\omega _{lm}$, which, from the symmetry of the problem [23], is given by

$$\begin{aligned} p(\omega _{lm})=\frac{1}{d_2D}; \end{aligned}$$

(41)

$p(\omega _{lm}|\hat{\rho }_{jk})$ is the conditional probability of obtaining the outcome $\omega _{lm}$ given that the prepared state was $\hat{\rho }_{jk}$. As we showed in Sect. 2, the state of system 2 is perfectly discriminated by a projective measurement onto the computational basis $\{|m\rangle \langle m|\}_{m=0}^{d_2-1}$. However, for discriminating the states of system 1, we need, in general, to implement an N-outcome generalized measurement $\{\hat{\varPi }_l^{{\mathcal {S}}}\}_{l=0}^{N-1}$, with $N\ge D$, where ${{\mathcal {S}}}$ stands for the strategy to be adopted. Therefore, we obtain

$$\begin{aligned} p(\omega _{lm}|\hat{\rho }_{jk})= & {} {\mathrm{Tr}}(\hat{\rho }_{jk}\hat{\varPi }_{lm})\nonumber \\= & {} {\mathrm{Tr}}\left[ \left( |\alpha _j\rangle \langle \alpha _j|\otimes |k\rangle \langle k|\right) \left( \hat{\varPi }_l^{{\mathcal {S}}}\otimes |m\rangle \langle m|\right) \right] \nonumber \\= & {} {\mathrm{Tr}}(\hat{\rho }_{j}\hat{\varPi }_l^{{\mathcal {S}}})\delta _{km}. \end{aligned}$$

(42)

Replacing these results in Eq. (40), the conditional entropy will be given by

$$\begin{aligned}{}[H(M|R)]^{{\mathcal {S}}}= & {} -\frac{1}{d_2D}\sum _{j=0}^{D-1}\sum _{l=0}^{N-1}\sum _{k,m=0}^{d_2-1} {\mathrm{Tr}}(\hat{\rho }_{j}\hat{\varPi }_l^{{\mathcal {S}}})\delta _{km}\log _2{\mathrm{Tr}}(\hat{\rho }_{j}\hat{\varPi }_l^{{\mathcal {S}}}) \nonumber \\= & {} -\frac{1}{D}\sum _{j=0}^{D-1}\sum _{l=0}^{N-1} {\mathrm{Tr}}(\hat{\rho }_{j}\hat{\varPi }_l^{{\mathcal {S}}})\log _2{\mathrm{Tr}}(\hat{\rho }_{j}\hat{\varPi }_l^{{\mathcal {S}}}) \nonumber \\= & {} [H(M_1|R_1)]^{{\mathcal {S}}}, \end{aligned}$$

(43)

where $[H(M_1|R_1)]^{{\mathcal {S}}}$ is the conditional entropy associated with the encoding/decoding process for the part of the message encoded in system 1. Using the results of Eqs. (37) and (43) in Eq. (36), we obtain the mutual information of Eq. (11), which ends the proof.